Introduction to shock & vibration

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Introduction to shock & vibration gives an introduction to vibration through a description of the most common mechanical parameters leading to the behaviour of simple mass spring systems. Furthermore the different types of signals and their description is treated and the conversion between the different parameters is described mathematically and graphically. Finally the measurement units are defined.

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Nội dung Text: Introduction to shock & vibration

,QWURGXFWLRQ WR 6KRFN 9LEUDWLRQ l Definitions l What is Vibration l Mechanical Parameters l Mass-spring Systems l How to Quantify Vibration l Signal Types l Time Signal Descriptors l Conversions: Acceleration, Velocity, Displacement l Units BA 7674-12, 1 $EVWUDFW The lecture gives an introduction to vibration through a description of the most common mechanical parameters leading to the behaviour of simple mass-spring systems. Furthermore the different types of signals and their description is treated and the conversion between the different parameters is described mathematically and graphically. Finally the measurement units are defined. Copyright© 1998 Brüel & Kjær Sound and Vibration Measurement A/S All Rights Reserved /(&785(127( English BA 7674-12
'HILQLWLRQV 9LEUDWLRQ is an oscillation wherein the quantity is a parameter defining the motion of a mechanical system 2VFLOODWLRQ is the variation, usually with time, of the magnitude of a quantity with respect to a specified reference when the magnitude is alternately greater and smaller than the reference BA 7674-12, 2 'HILQLWLRQV It is practical to know more precisely what we are going to talk about. These definitions are adapted from the “Shock and Vibration Handbook” by Harris and Crede (see literature list). Page 2
9LEUDWLRQ,Q(YHU\GD\/LIH UH WUIJNMMJ IONGVGã NJ KDSZZ SRGMGH R BA 7674-12, 3 :KDWLV9LEUDWLRQ" Vibration is mechanical oscillation about a reference position. Vibration is an everyday phenomenon, we meet it in our homes, during transport and at work. Vibration is often a destructive and annoying side effect of a useful process, but is sometimes generated intentionally to perform a task. 9LEUDWLRQRIPDFKLQHV Vibration is a result of dynamic forces in machines which have moving parts and in structures which are connected to the machine. Different parts of the machine will vibrate with various frequencies and amplitudes. Vibration causes wear and fatigue. It is often responsible for the ultimate breakdown of the machine. Page 3
8VHIXO9LEUDWLRQ BA 7674-12, 4 8VHIXO$SSOLFDWLRQRI9LEUDWLRQ Vibration is generated intentionally in component feeders, concrete compactors, ultrasonic cleaning baths and pile drivers, for example. Vibration testing machines impart vibration to objects in order to test their resistance and function in vibratory environments. Page 4
0HFKDQLFDO3DUDPHWHUVDQG&RPSRQHQWV Displacement Velocity Acceleration m d v a k c m F=k×d F=c×v F=m×a BA 7674-12, 5 0HFKDQLFDO3DUDPHWHUV Before going into a discussion about vibration measurement and analysis, we will examine the basic mechanical parameters and components and how they interact. All mechanical systems contain the three basic components: spring, damper, and mass. When each of these in turn is exposed to a constant force they react with a constant displacement, a constant velocity and a constant acceleration respectively. Page 5
6LPSOHVW)RUPRI9LEUDWLQJ6\VWHP Displacement Displacement d = D sinωnt D Time T Frequency 1 T m Period, Tn in [sec] k 1 Frequency, fn= T in [Hz = 1/sec] n k ωn= 2 π fn = m BA 7674-12, 6 0DVVDQG6SULQJ Once a (theoretical) system of a mass and a spring is set in motion it will continue this motion with constant frequency and amplitude. The system is said to oscillate with a sinusoidal waveform. 7KH6LQH&XUYH The sine curve which emerges when a mass and a spring oscillate can be described by its amplitude (D) and period (T). Frequency is defined as the number of cycles per second and is equal to the reciprocal of the period. By multiplying the frequency by 2π the angular frequency is obtained, which is again proportional to the square root of spring constant k divided by mass m. The frequency of oscillation is called the natural frequency fn. The whole sine wave can be described by the formula d = Dsin ωnt, where d = instantaneous displacement and D = peak displacement. Page 6
)UHH9LEUDWLRQ D Energy transfer between Kinetic and Potential Energy (assuming no damping) ∆ Kinetic Energy = - ∆ Potential Energy 1/2 m V2 = 1/2 k D2 , and V = (2πfn)D 1/2 m (2πfn)2 D2 = 1/2 k D2 1 k fn = 2π m BA 7674-12, 7 )UHHXQGDPSHGYLEUDWLRQ When a free undamped mass-spring system is set into oscillation the added energy is constant, but changes form from kinetic to potential during the motion. At maximum displacement the velocity and therefore also the kinetic energy is zero, while the potential energy is 1/2kD2. At the equilibrium position the potential energy is zero and the kinetic energy is maximum at 1/2mV2. For the sinusoidal motion d = D sinωnt we can also find the velocity by differentiating: d(Dsinω n t) v= = ω nDcosω n t = Vcosω n t dt and thereby find V = 2πfnD. Using energy conservation laws we then get the natural resonance frequency 1 k fn = 2π m Page 7
0DVVDQG6SULQJ time m1 k ωn = 2πIQ = m + m1 m Increasing mass reduces frequency BA 7674-12, 8 ,QFUHDVHRI0DVV An increase in the mass of a vibrating system causes an increase in period i.e. a decrease in frequency. Page 8
0DVV6SULQJDQG'DPSHU time Increasing damping reduces the amplitude m k c1 + c2 BA 7674-12, 9 0DVV6SULQJDQG'DPSHU When a damper is added to the system it results in a decrease in amplitude with time. The frequency of oscillation known as the damped natural frequency is constant and almost the same as the natural frequency. The damped natural frequency decreases slightly for an increase in damping. Page 9
)RUFHG9LEUDWLRQ H P Displacement 7L m dm k c Frequency dF Magnitude F dm = df Phase Frequency +90° 0° -90° Frequency BA 7674-12, 10 )RUFHG9LEUDWLRQ If an external sinusoidal force is applied to the system, the system will follow the force, which means that the movement of the system will have the same frequency as the external force. There might, however, be a difference in amplitude (and phase) as shown in the diagram. For frequencies below its natural frequency, the amplitude of the vibrating system will increase as the frequency is increased, a maximum being reached at the natural frequency. If there was no damping in the system (c = 0), the amplitude would approach infinity. If the frequency of the external force is increased the frequency of the spring/mass/damper system will increase to the same value, but the amplitude (and the phase) will change in accordance with the curves in the diagrams. Page 10
5HVSRQVHV&RPELQH Magnitude 1+2 d1+ d2 2 1 Frequency m d1 Phase Frequency dF 0° F 1 2 -90° 1+2 -180° BA 7674-12, 11 &RPELQHGUHVSRQVHV When considering real mechanical systems they will normally be more complex than the previous models. A simple example of two masses/springs/dampers is shown here. In this system we will see the responses combined, and its frequency response function shows two resonance peaks corresponding to the two masse/spring/damper systems. Page 11
5HVSRQVH0RGHOV Single Degree of Freedom Multi Degree of Freedom SDOF MDOF F Magnitude Magnitude F f0 Frequency Frequency BA 7674-12, 12 6LQJOH'HJUHHRI)UHHGRP6\VWHP A system consisting of only one mass, one spring and one damper is called a single degree of freedom system (if it can move in one direction only; if this system also can move sideways it is said to have two degrees of freedom and the discussion in the following diagram will apply). The phase is normally ignored in general vibration measurements, but it is very important when system analysis is made. 0XOWL'HJUHHRI)UHHGRP6\VWHP If the mechanical system consist of a number of interacting masses, springs and dampers or it can move in more than one direction, it is called a multi- degree of freedom system and the frequency spectrum will have one peak for each degree of freedom. Most systems are multi-degree of freedom systems, although it can often be difficult to separate the different mechanical components and even more difficult to design models as simple as this! Page 12
³5HDOZRUOG´5HVSRQVH Magnitude Rotor Bearing Bearing Foundation Frequency BA 7674-12, 13 5HDOZRUOG5HVSRQVH In most cases even simple systems are to be considered multi degree of freedom systems as illustrated here by a simple rotor in a couple of bearings. Page 13
)RUFHVDQG9LEUDWLRQ System Input Forces + Response = Vibration (Mobility) + = Frequency Frequency Frequency )RUFHVFDXVHGE\ 6WUXFWXUDO 9LEUDWLRQ l Imbalance 3DUDPHWHUV: 3DUDPHWHUV l Shock l Mass l Acceleration l Friction l Stiffness l Velocity l Acoustic l Damping l Displacement BA 7674-12, 14 891875 )RUFHVDQGYLEUDWLRQ A system will respond to an input force with a certain motion, depending on what we call the mobility of the system. Knowing the force and the mobility permits us to calculate the vibration. Modal analysis or other methods are used to model systems. Once the model is created we can calculate its mobility for a force input at a certain point, and thereby predict vibration at different locations. Such models can also in some cases be used to calculate the load on the structure to predict failure. Page 14
:K\'R:H0HDVXUH9LEUDWLRQ" l To verify that frequencies and amplitudes do not exceed the material limits (e.g. as described by the Wöhler curves) l To avoid excitation of resonances in certain parts of a machine l To be able to dampen or isolate vibration sources l To make conditional maintenance on machines l To construct or verify computer models of structures (system analysis) BA 7674-12, 15 :K\PHDVXUHYLEUDWLRQ" A number of reasons are listed here. The Wöhler curve is a curve describing the stress level up to which a structure can be loaded a certain number of times (endurance strength). At high stresses the load can only be carried a few times, but reducing the stress increases the number of cycles to failure. For most metals there exists an endurance limit for which the endurance becomes infinite. This stress level is very important, and it is often found by subjecting the object to 10.000.000 cycles of stress, based on the experience that this number is sufficient to reach the endurance limit. Page 15
+RZGR:H4XDQWLI\9LEUDWLRQ" l We make a measurement l We analyse the results (levels and frequencies) In order to make the analysis, we must first talk about the types of vibration signals we might encounter and how we measure these signals BA 7674-12, 16 Page 16
7\SHVRI6LJQDOV Stationary signals Non-stationary signals Deterministic Random Continuous Transient BA 7674-12, 17 6LJQDOV Basically a distinction between Stationary Signals and Non-stationary Signals has to be made. Stationary Signals can again be divided into Deterministic Signals and Random Signals, and Non-stationary Signals into Continuous and Transient signals. Stationary deterministic signals are made up entirely of sinusoidal components at discrete frequencies. Random signals are characterised by being signals where the instantaneous value cannot be predicted, but where the values can be characterised by a certain probability density function i.e. we can measure its average value. Random signals have a frequency spectrum which is continuously distributed with frequency. The continuous non-stationary signal has some similarities with both transient and stationary signals. During analysis continuous non-stationary signals should normally be treated as random signals or separated into the individual transient and treated as transients. Transient signals are defined as signals which commence and finish at a constant level, normally zero, within the analysis time. Page 17
'HWHUPLQLVWLF6LJQDOV B C Amplitude Amplitude E CD B A A Time Frequency E Vibration D BA 7674-12, 18 'HWHUPLQLVWLF6LJQDOV The vibration signal from a gearbox could look like the one shown here. In the frequency domain this signal will give rise to a number of separate peaks (discrete frequency components) which through knowledge of the number of teeth on the gearwheels and their speed can be related back to particular parts of the system. The signal here is called deterministic, since the instantaneous value of the signal is predictable at all points in time. 7KHUROHRIIUHTXHQF\DQDO\VLV The frequency spectrum gives in many cases a detailed information about the signal sources which cannot be obtained from the time signal. The example shows measurement and frequency analysis of the vibration signal measured on a gearbox. The frequency spectrum gives information on the vibration level caused by rotating parts and tooth meshing. It hereby becomes a valuable aid in locating sources of increased (undesirable) vibration from these and other sources. Page 18
'HWHUPLQLVWLF6LJQDOVDQG+DUPRQLFV Time f1 2f1 Frequency Time f1 Frequency Time 2f1 Frequency BA 7674-12, 19 9LEUDWLRQ6LJQDOV The motion of a mechanical system can consist of a single component at a single frequency as with the system described in one of the previous examples; (a tuning fork is another example) or it can consist of several components occurring at different frequencies simultaneously, as for example with the piston motion of an internal combustion engine. The motion signal is here split up into its separate components both in the time domain and in the frequency domain. Page 19
+DUPRQLFV Generator Time f0 3f1 5f1 Frequency Time f0 2f0 3f0 4f0 5f0 Frequency BA 7674-12, 20 +DUPRQLFV Many non-sinusoidal signals can be separated into a number of harmonically related sinusoids. Two examples are given. The harmonic components are always referred to the fundamental frequency to which they are related. Page 20